mean field limit for coulomb-type flowsserfaty/revised-version-2.pdf · 2020. 3. 28. · mean field...

MEAN FIELD LIMIT FOR COULOMB-TYPE FLOWS

SYLVIA SERFATY, APPENDIX WITH MITIA DUERINCKX

Abstract. We establish the mean-field convergence for systems of points evolving alongthe gradient flow of their interaction energy when the interaction is the Coulomb potentialor a super-coulombic Riesz potential, for the first time in arbitrary dimension. The proofis based on a modulated energy method using a Coulomb or Riesz distance, assumes thatthe solutions of the limiting equation are regular enough and exploits a weak-strong stabilityproperty for them. The method can handle the addition of a regular interaction kernel, andapplies also to conservative and mixed flows. In the appendix, it is also adapted to prove themean-field convergence of the solutions to Newton’s law with Coulomb or Riesz interactionin the monokinetic case to solutions of an Euler-Poisson type system.

1. Introduction

The derivation of effective equations for classical interacting many body systems is animportant question in mathematical physics. Within it, one of the most important prob-lems, fundamental for plasma physics, is the derivation of the Vlasov-Poisson equation fromNewton’s law for N particles i.e. a system of the form

(1.1)

xi = vi

vi = 1N

∑i 6=j

K(xi, xj), i = 1, . . . , N

xi(0) = x0i vi(0) = v0

i

where the pair interaction K derives from the Coulomb potential, and is still open in its fullgenerality. The large N or mean-field limit of first order systems of the form

(1.2)

xi = 1

N

∑i 6=j

K(xi, xj), i = 1, . . . , N

xi(0) = x0i

is also a natural and interesting question. Such systems can model interacting particles inphysics (for instance the point vortex system in two-dimensional fluids), from a numericalpoint of view they correspond to particle approximations of PDEs, or in the case of gradientflows can serve to approximate their equilibrium states. Motivations extend to biologicaland sociological sciences, including phenomena of flocking, swarming and aggregation (see forinstance [CCH]) and the analysis of large neural networks in biology [BFT] and in machinelearning [MMn,RVE,BaCh].

In the above problems, the cases where K is regular are well understood, while in contrastthose where K is singular are the most difficult and least understood. In this paper we willbe particularly focusing on the case where K corresponds to the Coulomb interaction (andsome generalizations), arguably the most important case for physics. For further details onthe general mathematical aspects of mean-field limits, we refer to the reviews [Go,Jab].

1

2 SYLVIA SERFATY, APPENDIX WITH MITIA DUERINCKX

The rest of the introduction is structured as follows: in Section 1.1 we introduce the exactequations that we will study as well as their limiting effective evolution equations, and describethe state of the art on such questions. In Section 1.2 we state the main result, in Section 1.3we comment on the assumptions, and in Section 1.4 we give an extended proof outline forthe Coulomb case. Finally, Section 1.5 is an added section explaining how to treat the caseof gradient flow evolutions with thermal noise.

1.1. Problem and background. In this paper we consider specifically systems with Coulomb,logarithmic or Riesz interaction with kernel deriving from

(1.3) g(x) = |x|−s max(d− 2, 0) ≤ s < d for any d ≥ 1

or

(1.4) g(x) = − log |x| for d = 1 or 2,

where d is the dimension. In the case (1.3) with s = d − 2 and d ≥ 3, or (1.4) and d = 2, gis exactly (a multiple of) the Coulomb kernel. In the other cases of (1.3) it is called a Rieszkernel. In some cases, we may add to the interaction force a regular part denoted F.

We will consider first order dynamics of the form

(1.5)

xi = − 1

N

∇xiHN (x1, . . . , xN ) +∑j 6=i

F(xi − xj)

, i = 1, . . . , N

xi(0) = x0i

or conservative evolutions of the form

(1.6)

xi = − 1N

J∇xiHN (x1, . . . , xN ), i = 1, . . . , N

xi(0) = x0i

where J is an antisymmetric matrix, the points xi evolve in the whole space Rd and theirenergy HN is given by

(1.7) HN (x1, . . . , xN ) =∑i 6=j

g(xi − xj),

with g as above. The map F : Rd → Rd is the additional interaction force that we can addin the dissipative case to illustrate the robustness of the method, we will require it to enjoysome Sobolev and Holder regularity.

Mixed flows of the form

(1.8) xi = − 1N

(αI + βJ)

∇xiHN (x1, . . . , xN ) +∑j 6=i

F(xi − xj)

α > 0

can be treated with exactly the same proof, so can the same dynamics (1.5), (1.6) or (1.8)with an additional forcing 1

N

∑Ni=1 V(xi) with V Lipschitz. These generalizations are left to

the reader, see in particular Remark 2.2.Studying the same evolutions with added noise

(1.9) dxi = − 1N

∇xiHN (x1, . . . , xN ) +∑j 6=i

F(xi − xj)

dt+√

2θdWi,

MEAN FIELD LIMIT FOR COULOMB FLOWS 3

or

(1.10) dxi = − 1N

J∇xiHN (x1, . . . , xN )dt+√

2θdWi,

with dWi being N independent Brownian motions and θ ≥ 0 a temperature, is also veryinteresting and has motivations from Random Matrix Theory, it is done in particular in[JW1, JW2, BO, FHM, BCF, LLY] (see also references therein). We will comment further onthis at the end of the introduction in Section 1.5.

In the appendix, we consider the second order system corresponding to Newton’s law forthe energy HN

xi = vi,

vi = − 1N∇xiHN (x1, . . . , xN ),

xi(0) = x0i , vi(0) = v0

i ,

i = 1, . . . , N(1.11)

in the so-called monokinetic case. In the Coulomb case it is the true physics model forplasmas.

Consider the empirical measure

(1.12) µtN := 1N

N∑i=1

δxti

associated to a solution XtN := (xt1, . . . , xtN ) of the flow (1.5) or (1.6). If the points x0

i ,which themselves depend on N , are such that µ0

N converges to some regular measure µ0, thena formal derivation leads to expecting that for t > 0, µtN converges to the solution of theCauchy problem with initial data µ0 for the limiting evolution equation(1.13) ∂tµ = div ((∇g + F) ∗ µ)µ)in the dissipative case (1.5) or(1.14) ∂tµ = div (J∇(g ∗ µ)µ)in the conservative case (1.6), with ∗ denoting the usual convolution.

These equations should be understood in a weak sense. Equation (1.13) (with F = 0)is sometimes called the fractional porous medium equation. The two-dimensional Coulombversion also arises as a model for the dynamics of vortices in superconductors. The construc-tion of solutions, their regularity and basic properties, are addressed in [LZ,DZ,MZ,AS,SV]for the Coulomb case of (1.13), in [CSV, CV, XZ] for the case d − 2 < s < d of (1.13),and [De,Yu,Scho1] for the two-dimensional Coulomb case of (1.14).

Establishing the convergence of the empirical measures to solutions of the limiting equa-tions is nontrivial because of the nonlinear terms in the equation and the singularity of theinteraction g. In fact, because of the strength of the singularity, treating the case of Coulombinteractions in dimension d ≥ 3 (and even more so that of super-coulombic interactions) hadremained an open question for a long time, see for instance the introduction of [JW2] and thereview [Jab]. It was not even completely clear that the result was true without expressing itin some statistical sense (with respect to the initial data).

In [JW1, JW2], Jabin and Wang introduced a new approach for the related problem ofthe mean-field convergence of the solutions of Newton’s second order system of ODEs to theVlasov equation, which allowed them to treat all interactions kernels with bounded gradients,but still not Coulomb interaction. The same problem has been addressed in [BP,LP,La] with


results that still require a cutoff of the Coulomb interaction. Our method already allows tounlock the case of Coulomb interaction for monokinetic data, this is the topic of Appendix A.The non-monokinetic case is even much more challenging and remains open.

The previously known results on the problems we are addressing were the following:• In dimension 2, choosing (1.4) and J the rotation by π/2 in (1.6) corresponds to

the so-called point vortex system which is well-known in fluid mechanics (cf. forinstance [MP]), and its mean-field convergence to the Euler equation in vorticity form(1.14) was already established [Scho2]. His proof can be readapted to treat (1.5) aswell in that case. Results of similar nature were also obtained in [GHL].• Hauray [Hau] (see also [CCH]) treated the case of all sub-Coulombic interactions

(s < d− 2) for (a possibly higher-dimensional generalization of) (1.6), where particlescan have positive and negative charges and thus can attract as well as repel. Theproof, which relies on the stability in∞-Wasserstein distance of the limiting solution,cannot be adapted to s ≥ d− 2.• In dimension 1 and in the dissipative case (1.5), Carrillo-Ferreira-Precioso and Berman-

Onnheim [CFP,BO] proved the unconditional convergence for all 0 < s < 1 using theframework of Wasserstein gradient flows but their method, based on the convexity ofthe interaction in dimension 1, does not extend to higher dimensions.• Duerinckx [Du], inspired by the modulated energy method of [Sy] for Ginzburg-Landau

equations (where vortices also interact like Coulomb particles in dimension 2), wasable to prove the result in the dissipative case (1.5) for d = 1 and d = 2 with s < 1,conditional to the regularity of the limiting solution, as we have here.

In this paper, we prove the mean field convergence for (1.5) and (1.6) in all the cases (1.3)and (1.4) in every dimension. This extends Duerinckx’s result, which involves overcomingserious difficulties, as described further below, and we add the possible additional interactionforce F in the dissipative case. We are limited to s < d and this is natural since for s ≥ dthe interaction kernel g is no longer integrable and the limiting equation is expected to be adifferent one.

As in [Du], our proof is a modulated energy argument inspired from [Sy], which is a way ofexploiting a weak-strong uniqueness principle for the limiting equation. As mentioned above,looking for a stability principle in some Wasserstein distance fails for the Coulomb singularity.Instead we use a distance which is built as a Coulomb (or Riesz) metric, associated to thenorm

(1.15) ‖µ‖2 =¨

g(x− y)dµ(x)dµ(y).

We are able to show by a Gronwall argument on this metric that the equations (1.13) and(1.14) satisfy a weak-strong uniqueness principle, and this can be translated into a proof ofstability and convergence to 0 of the norm of µtN − µt (if it is initially small, it remains smallfor all further times).

The proof is self-contained and quantitative. It does not require understanding any quali-tative property of the trajectories of the particles, such as for instance their minimal distancesalong the flow.

After this work was completed, Bresch, Jabin and Wang [BJW1] were able to beautifullyincorporate our modulated energy into the relative entropy method of [JW2], turning it into amodulated free energy, which is a physically very natural quantity. It allowed them to extend


the result of [JW2] to more singular interactions, including Coulomb. Seen from our point ofview, it allows to treat the cases with added noise (1.9) (but unfortunately not (1.10)). Thiscan be explained very succintly, we do it in Section 1.5 for the reader’s convenience.

1.2. Main result. Let XN denote (x1, . . . , xN ) and let us define for any probability measureµ,

(1.16) FN (XN , µ) =¨

Rd×Rd\4g(x− y)d

( N∑i=1

δxi −Nµ)(x)d

( N∑i=1

δxi −Nµ)(y)

where 4 denotes the diagonal in Rd × Rd. We choose for “modulated energy”

FN (XtN , µ

t)

where XtN = (xt1, . . . , xtN ) are the solutions to (1.5) or (1.6), and µt solves the expected

limiting PDE. The function FN is not positive, however it is bounded below (by −CN1+ sd in

the case (1.3), respectively −(Nd logN

)− CN in the case (1.4), see Corollary 3.5). It turns

out that also metrizes at least weak convergence, as described in Proposition 3.6. One maythus think of it as a good notion of distance from µtN to µt, more precisely 1

N2FN (XtN , µ

t) isa good distance.

Our main result is a Gronwall inequality on the time-derivative of FN (XtN , µ

t), whichimplies a quantitative rate of convergence of µtN to µt in that metric.

Throughout the paper, (·)+ denotes the positive part of a number. The parameter s refersto the exponent in (1.3) while in the logarithmic case (1.4) it is set to be 0. The spaceHm(Rd) is the homogeneous Sobolev space of functions u whose Fourier transform u satisfies|ξ|mu(ξ) ∈ L2(Rd).

Theorem 1. Assume that g is of the form (1.3) or (1.4). Assume that F ∈ Hd−s

2 (Rd) ∩C0,α(Rd) for some α > 0 and ∇F ∈ Lq(Rd) for some 1 ≤ q ≤ ∞. Assume (1.13), resp.(1.14), admits a solution µt such that, for some T > 0,(1.17){µt ∈ L∞([0, T ], L∞(Rd)), and ∇2g ∗ µt ∈ L∞([0, T ], L∞(Rd)) if s < d− 1µt ∈ L∞([0, T ], Cσ(Rd)) with σ > s− d + 1, and ∇2g ∗ µt ∈ L∞([0, T ], L∞(Rd)) if s ≥ d− 1.

Let XtN solve (1.5), respectively (1.6). Then there exist positive constants C1, C2 depending

only on the norms of µt controlled by (1.17) and those of F, and an exponent β < 2 dependingonly on d, s, α, σ, such that for every t ∈ [0, T ] we have

(1.18) FN (XtN , µ

t) ≤(FN (X0

N , µ0) + C1N

β)eC2t.

In particular, using the notation (1.12), if µ0N ⇀ µ0 and is such that

limN→∞

1N2FN (X0

N , µ0) = 0,

then the same is true for every t ∈ [0, T ] and

(1.19) µtN ⇀ µt

in the weak sense.


Establishing the convergence of the empirical measures is essentially equivalent to provingpropagation of molecular chaos (see [Go,HM,Jab] and references therein) which means showingthat if f0

N (x1, . . . , xN ) is the initial probability density of seeing particles at (x1, . . . , xN ) andif f0

N converges to some factorized state µ0⊗· · ·⊗µ0, then the k-point marginals f tN,k convergefor all time to (µt)⊗k. With Remark 3.7, our result implies a convergence of this type as well.

Let us point out that using a Fourier-based point of view on (1.16) Bresch-Jabin-Wangwere able (see [BJW1,BJW2]) to subsequently relax the assumptions on the interaction g: itdoes not need to be Coulomb or Riesz (a bit like with the added regular force in (1.5)) butmay contain a mildly singular attractive part, as long as a sufficiently strong repulsive partis still present.

1.3. Comments on the assumptions. Let us now comment on the regularity assumptionmade in (1.17). First of all, one can check (see Lemma 3.1) that the assumption (1.17) isimplied by

(1.20) µ ∈ L∞([0, T ], Cθ(Rd)) for some θ > s− d + 2,

which coincides with the assumption made in [Du] and is a bit stronger. This weakeningof the assumption allows to include for instance the case of measures which are (a regularfunction times) the characteristic function of some regular set, such as in the situation ofvortex patches for the Euler equation in vorticity form, corresponding to (1.6) in the two-dimensional logarithmic case. These vortex patches were first studied in [Ch2, BerCon, Si]where it was shown that if the patch initially has a C1,α boundary this remains the caseover time, and our second assumption that the velocity ∇g ∗ µt be Lipschitz holds as well(see also [BK]). It is not too difficult to check that in all dimensions this second conditionholds any time µ is Cσ with σ > 0 away from a finite number of C1,α hypersurfaces. Moregenerally, such situations with patches can be expected to naturally arise in all the Coulombcases. For instance, in the dissipative Coulomb case (1.5) with F = 0, in any dimension aself-similar solution in the form of (a constant multiple of) the characteristic function of anexpanding ball was exhibited in [SV] and shown to be an attractor of the dynamics. Forthe non-Coulomb dissipative cases, the corresponding self-similar solutions, called Barenblattsolutions, are of the form

t−d

2+s (a− bx2t−2

2+s )s−d+2

2+

as shown in [BIK,CV] (and this formula retrieves the solution of [SV] for s = d− 2).If the initial µ0 is sufficiently regular, the stronger assumption (1.20) is known to hold with

T = ∞ for the Coulomb case (see [LZ] where the proof works as well in higher dimensions),and it is known up to some T > 0 in the case (d − 2)+ < s ≤ d − 1 [XZ]. As for (1.14), toour knowledge the desired regularity is only known in dimension 2 for the Euler equation invorticity form (see [Wo,Ch2]), although the arguments of [XZ] written for the dissipative caseseem to also apply to the conservative one. Our convergence result thus holds in these cases,under the assumption that the limit µ0 of µ0

N is sufficiently regular and that FN (X0N , µ

0) =o(N2). Note that, as shown in [Du], the latter is implied by the convergence of the initialenergy

limN→∞

1N2

∑i 6=j

g(x0i − x0

j ) =¨

Rd×Rdg(x− y)dµ0(x)dµ0(y)


which can be viewed as a well-preparedness condition. For any regular enough µ0, one mayfor instance build initial conditions satisfying this assumption (and something even stronger)by the construction in [PS].

For d − 1 < s < d, even the local in time propagation of regularity of solutions of (1.13)remains an open problem. Note that the uniqueness of regular enough solutions is alwaysimplied by the weak-strong stability argument we use, reproduced in Section 1.4 below.

Requiring some regularity of the solutions to the limiting equation for establishing con-vergence with relative entropy / modulated entropy / modulated energy methods is fairlycommon: the same situation appears for instance in [JW1, JW2] or in the derivation of theEuler equations from the Boltzmann equation via the modulated entropy method, see [SR]and references therein.

1.4. The method. As mentioned, our method exploits a weak-strong uniqueness principlefor the solutions of (1.13), resp. (1.14), which is exactly the same as [Du, Lemma 2.1, Lemma2.2] (and can be easily readapted to the conservative case) and states that if µt1 and µt2 aretwo L∞ solutions to (1.13) such that ∇2(g ∗ µ2) ∈ L1([0, T ], L∞), we have

(1.21)¨

Rd×Rdg(x− y)d(µt1 − µt2)(x)d(µt1 − µt2)(y)

≤ eC´ t

0 ‖∇2(g∗µs2)‖ds

¨Rd×Rd

g(x− y)d(µ01 − µ0

2)(x)d(µ01 − µ0

2)(y).

But the Coulomb or Riesz energy (1.15) is nothing else than the fractional Sobolev H−α normof µ with α = d−s

2 , hence this is a good metric of convergence and implies the weak-stronguniqueness property.

A crucial ingredient is the use of the stress-energy (or energy-momentum) tensor whichnaturally appears when taking the inner variations of the energy (1.15), i.e. computingddt |t=0‖µ ◦ (I + tψ)‖2 (this is standard in the calculus of variations, see for instance [He, Sec.1.3.2]). To explain further, let us restrict for now to the Coulomb case, and set hµ = g ∗ µ,the Coulomb potential generated by µ. In that case, we have(1.22) −∆hµ = cdµ

for some constant cd depending only on d. The first key is to reexpress the Coulomb energy(1.15) as a single integral in hµ, more precisely we easily find via an integration by parts thatif µ is a measure with

´dµ = 0,

(1.23)¨

Rd×Rdg(x− y)dµ(x)dµ(y) =

ˆRdhµdµ = − 1

cd

ˆRdhµ∆hµ = 1

cd

ˆRd|∇hµ|2.

The stress-energy tensor is then defined as the d× d tensor with coefficients(1.24) [hµ, hµ]ij = 2∂ihµ∂jhµ − |∇hµ|2δij ,with δij the Kronecker symbol. We may compute that if µ is regular enough(1.25) div [hµ, hµ] = 2∆hµ∇hµ = −2cdµ∇hµ.(Here the divergence is a vector with entries equal to the divergence of each row of [hµ, hµ].)We thus see how this stress-energy tensor allows to give a weak meaning to the productµ∇hµ = µ∇g ∗ µ, with [hµ, hµ] well-defined in energy space and pointwise controlled by|∇hµ|2, which can by the way serve to give a notion of weak solutions of the equation in theenergy space (as in [De, LZ]). Note that in dimension 2, it is known since [De] that even


though [hµ, hµ] is nonlinear, it is stable under weak limits in energy space provided µ has asign, but this fact does not extend to higher dimension.

Let us now present the short proof of (1.21) as it will be a model for the main proof.We focus on the dissipative case (the conservative one is an easy adaptation) and still theCoulomb case for simplicity with no additional interaction F (when present, the additionalterms can be absorbed thanks to the dissipation). Let µ1 and µ2 be two solutions to (1.13)and hi = g ∗ µi the associated potentials, which solve (1.22). Let us compute

(1.26) ∂t

ˆRd|∇(h1 − h2)|2 = 2cd

ˆRd

(h1 − h2)∂t(µ1 − µ2)

= 2cd

ˆRd

(h1 − h2)div (µ1∇h1 − µ2∇h2)

= −2cd

ˆRd

(∇h1 −∇h2) · (µ1∇h1 − µ2∇h2)

= −2cd

ˆRd|∇(h1 − h2)|2µ1 − 2cd

ˆRd∇h2 · ∇(h1 − h2)(µ1 − µ2)

In the right-hand side, we recognize from (1.25) the divergence of the stress-energy tensor[h1 − h2, h1 − h2], hence

∂t

ˆRd|∇(h1 − h2)|2 ≤

ˆRd∇h2 · div [h1 − h2, h1 − h2]

so if ∇2h2 is bounded, we may integrate by parts the right-hand side and bound it pointwiseby

‖∇2h2‖L∞ˆRd|[h1 − h2, h1 − h2]| ≤ 2‖∇2h2‖L∞

ˆRd|∇(h1 − h2)|2,

and the claimed result follows by Gronwall’s lemma.In the Riesz case, the Riesz potential hµ = g∗µ is no longer the solution to a local equation,

and to find a replacement to (1.22)–(1.25), we use an extension procedure as popularizedby [CaffSi] in order to obtain a local integral in hµ in the extended space Rd+1.

In the discrete case of the original ODE system, all the above integrals are singular andthis constitutes the main difficulty overcome in this paper. In place of the second term in theright-hand side of (1.26), we then have to control a term which by symmetry can be writtenin the form

(1.27)¨

Rd×Rd\4(∇hµ1(x)−∇hµ2(y)) · ∇g(x− y)d(µ1 − µ2)(x)d(µ1 − µ2)(y)

where 4 denotes the diagonal, µ1 is the limiting measure µt and µ2 is the discrete empiricalmeasure µtN . Such terms are well known (see for instance [Scho2]), and create the maindifficulty due to the singularity of g. When removing the diagonal, the positivity manifestedby (1.23) is in effect lost. We are however able to prove the following crucial functionalinequality.

Proposition 1.1. Assume that µ is a probability density, with µ ∈ Cσ(Rd) with σ > s−d + 1if s ≥ d − 1; respectively µ ∈ L∞(Rd) or µ ∈ Cσ(Rd) with σ > 0 if s < d − 1. For any


XN ∈ (Rd)N and any Lipschitz map ψ : Rd → Rd, we have

(1.28)∣∣∣∣∣¨4c

(ψ(x)− ψ(y)) · ∇g(x− y)d(N∑i=1

δxi −Nµ)(x)d(N∑i=1

δxi −Nµ)(y)∣∣∣∣∣

≤ C‖∇ψ‖L∞(FN (XN , µ) + (1 + ‖µ‖L∞)N2− d−s

d(d+1) + 2(N

d logN)

1(1.4)

)+ C min

(‖ψ‖L∞‖µ‖L∞N1+ s+1

d + ‖∇ψ‖L∞‖µ‖L∞N1+ sd , ‖ψ‖W 1,∞‖µ‖CσN1+ s+1−σ

d)

+ C

{‖∇ψ‖L∞(1 + ‖µ‖Cσ)N2− 1

d if s ≥ d− 1‖∇ψ‖L∞(1 + ‖µ‖L∞)N2− 1

d if s < d− 1,where C depends only on s, d.

The right-hand side should be read as Cψ,µ,s,d(FN (XN , µ) + Nβ) for some β < 2. Thisinequality is the main novelty of the paper. Even though the term (ψ(x)− ψ(y)) · ∇g(x− y)has a singularity of same order as g(x − y) near the diagonal, the inequality is not at allobvious due to the lack of positivity of the integrand and its proof is rendered difficult by thehandling of the removed diagonal terms. Note that in [GP] Golse and Paul were able to treatthe mean field limit for the quantum Coulomb dynamics, relying on this inequality.

To give more insight into the proof of this proposition, we need to discuss the electricrepresentation of the modulated energy FN , again restricting to the Coulomb case. For thatwe introduce the electric potential

HµN [XN ] = g ∗

(N∑i=1

δxi −Nµ).

Arguing as in (1.22)–(1.23) we would like to rewrite FN (XN ) as´Rd |∇Hµ

N [XN ]|2. This is notquite correct due to the divergence of Hµ

N at the points xi. This can however be corrected byusing mollified Dirac masses and setting for any ~η = (η1, . . . , ηN ) ∈ RN ,

HµN,~η = g ∗

(N∑i=1

δ(ηi)xi −Nµ

)

where we dropped the [XN ] in the notation and let δ(η)x denote the uniform measure of mass

1 on ∂B(x, η). This effectively truncates HµN at scale ηi around each xi, i.e. a scale which

depends on each point. Reinserting the diagonal terms, it is then not too difficult to checkthat

FN (XN , µ) = limηi→0

¨g(x− y)d

(N∑i=1

δ(ηi)xi −Nµ

)(x)d

(N∑i=1

δ(ηi)xi −Nµ

)(y)

−N∑i=1

¨g(x− y)dδ(ηi)

xi (x)dδ(ηi)xi (y)

= 1cd

limηi→0

(ˆRd|∇Hµ

N,~η|2 − cd

N∑i=1

g(ηi)).

This effectively gives a renormalized meaning to the identity (1.23) in this setting.The idea of expressing the interaction energy as a local integral in Hµ

N and its renor-malization procedure were previously used in the study of Coulomb and Riesz energies


in [RS, PS, LS1, LS2], but it was not clear how to adapt these ideas to control (1.27): infact in [Du] this was dealt with by a ball-construction procedure inspired from the analysisof Ginzburg-Landau vortices, which led to nonoptimal estimates and to the restriction to thedissipative case only and to s < 1 and d ≤ 2.

In fact, we can say better, and this is where we depart significantly from previous works, byexploiting the fact that the expression

´Rd |∇Hµ

N,~η|2 − cd

∑Ni=1 g(ηi) is essentially decreasing

with respect to ηi and constant for ηi small enough that the B(xi, ηi)’s are disjoint, seeProposition 3.3. More precisely, we may set ri to be 1/4 of the minimal distance from xi toall other points, and for ηi ≤ ri, we have the equality (without limits)

(1.29) FN (XN , µ) = 1cd

(ˆRd|∇Hµ

N,~η|2 − cd

N∑i=1

g(ηi))

+ explicit negligible terms.

In addition, an observation made in this paper for the first time is that when choosing preciselyηi = ri, the potentially large terms

´Rd |∇Hµ

N,~r|2 and cd

∑Ni=1 g(ri) are separately controlled

by CFN (plus good terms) and conversely, see Corollary 3.4. It now suffices to control theleft-hand side of (1.28) by

´Rd |∇Hµ

N,~r|2. Let us emphasize that this choice of truncation ri

that depends on the point is (up to constants) the only one which is at the same time largeenough so that

´Rd |∇Hµ

N,~η|2 is directly controlled by FN and small enough that the equality

(1.29) holds. In other words, since we do not have any bound from below on the distancebetween the points, a point-dependent truncation radius is crucial. As a side note, the ideaof using truncations for proving mean-field limits is common, however what is usually doneis to truncate the interaction g itself (at lengthscales possibly depending on N), and try totake limits in the resulting flow. What we do here is very different: we do not modify theinteraction but desingularize the charges themselves as an intermediate step to control thesingular terms.

In order to bound the left-hand side of (1.28), the key is then to interpret it as well as asingle integral in terms of ∇Hµ

N in a suitable “renormalized” way, more precisely in terms ofthe stress tensor associated to ∇Hµ

N,~η, for ηi → 0. The quantity then obtained is this timenot monotone in ηi, however, by carefully studying its variations in ηi (this is the hardestpart of the proof), we are able to control it by the (integral of the) stress tensor associatedto ∇HN,~r, which can in turn be bounded by

´|∇Hµ

N,~r|2, and we conclude thanks to the fact

that this is controlled by CFN .

1.5. The modulated free energy for the case with noise. In this subsection, we explainthe modulated free energy method of [BJW1], which is posterior to the first version of thiswork and allows to treat the case of (1.9). In this case, the mean-field limit inherits an addedLaplacian:

(1.30) ∂tµ = div ((∇g + F) ∗ µ)µ) + θ∆µ.

Consider fN (x1, . . . , xN ) a symmetric probability density on configurations in (Rd)N , andlet us again abbreviate (x1, . . . , xN ) by XN . Let us introduce the relative entropy 1

HN (fN |µ⊗N ) := N

ˆRdN

fN log fNµ⊗N

dXN

1Note that we take N2 times the usual relative entropy, because all our quantities are N2 times standardones.


It is of course a way of measuring how close the distribution fN is to µ⊗N . Then consider

KN (fN , µ) =ˆRdN

fN (XN )FN (XN , µ)dXN

the expectation of our modulated energy FN with respect to fN . Bresch-Jabin-Wang intro-duce the modulated free energy(1.31) Fθ(fN , µ) = θHN (fN |µ⊗N ) +KN (fN , µ).It has exactly the structure of a free energy in statistical physics, i.e. of the form energyplus temperature times entropy, and when temperature vanishes and fN concentrates on oneconfiguration, it coincides with the regular modulated energy.

Consider now f tN corresponding to the probability density of particles following the flow(1.9), then by Ito calculus f tN solves a Liouville or Kolmogorov equation

(1.32) ∂tftN =

N∑i=1

div i

1N

∑i 6=j∇iHN (xi − xj)f tN (XN )

+ θN∑i=1

∆iftN .

Their crucial observation is that when combining the relative entropy and the modulatedenergy in exactly the way of (1.31) and differentiating in time Fθ(f tN , µt) for µt a solutionto the mean-field limit (1.30) and f tN a solution to the Liouville equation (1.32), the newand problematic terms arising in ∂tHN (f tN , µt) from the presence of the noise (which were anobstacle to treat the Coulomb case in [JW2]) exactly cancel with the new terms arising in∂tKN (f tN , µt) (this does not happen in the conservative case of (1.10) though). This allowsthem to obtain the following crucial identity:

(1.33) d

dtFθ(f tN , µt)

≤ −ˆ (¨

4c∇g(x− y) · (ψt(x)− ψt(y))d(

N∑i=1

δxi −Nµt)(x)d(N∑i=1

δxi −Nµt)(y))df tN (XN )

with this timeψt = ∇hµt + θ

∇µt

µt.

Once this identity is observed, Proposition 1.1 directly applies (if µt is assumed regularenough) and yields

d

dtFθ(f tN , µt) ≤ CKN (f tN , µt) + o(N2) ≤ CFθ(f tN , µt) + o(N2)

which allows to directly conclude via Gronwall’s lemma the mean-field convergence in thecase with noise.

The rest of the paper is organized as follows: we start by deriving the main result assumingthe result of Proposition 1.1. In Section we present the details of the electric formulation inthe general Riesz case and prove the main properties of the modulated energy (monotonicity,bound from below, coerciveness). We conclude in Section 4 with the discussion of the stress-energy tensor and the proof of Proposition 1.1. The paper finishes with Appendix A on thederivation of the Vlasov-Poisson system in the monokinetic case.

Acknowledgments: I would like to thank Mitia Duerinckx for his careful reading andhelpful suggestions and Pierre-Emmanuel Jabin for useful discussions on the work [BJW1,BJW2]. This research was supported by the NSF grant DMS-1700278.


2. Main proof

In all the paper, we will use the notation 1(1.4) to indicate a term which is only presentin the logarithmic cases (1.4) and 1s<d−1 for a term which is present only if s < d − 1.Below, the principal value (which may be omitted for s < d− 1) is defined by P.V.

´Rd\{x} :=

limr→0´Rd\B(x,r) .

Differentiating from formula (1.16), we have

Lemma 2.1. If XtN is a solution of (1.5), then

(2.1) ∂tFN (XtN , µ

t) = −2N2ˆRd

∣∣∣∣∣P.V.ˆRd\{x}

∇g(x− y)d(µtN − µt)(y)∣∣∣∣∣2

dµtN (x)

+ 2N2ˆRd

F ∗ (µt − µtN ) · ∇(hµtN − hµt)dµtN

−N2¨

Rd×Rd\4

((∇hµt + F ∗ µt)(x)− (∇hµt + F ∗ µt)(y)

)·∇g(x−y)d(µtN−µt)(x)d(µtN−µt)(y).

If XtN is a solution of (1.6), then

(2.2)

∂tFN (XtN , µ

t) = −N2¨

Rd×Rd\4J(∇hµt(x)−∇hµt(y)

)·∇g(x−y)(µtN−µt)(x)(µtN−µt)(y).

Remark 2.2. In the case of an added term 1N

∑Ni=1 V(xi) in the evolutions, one obtains an

additional term

−2N2¨

Rd×Rd\4(V(x)− V(y)) · ∇g(x− y)d(µtN − µt)(x)d(µtN − µt)(y)

which can be handled like the others using Proposition 1.1 if V is globally Lipschitz.

Proof. We note that if s ≥ d− 1, ∇g is not integrable near 0, so ∇g ∗µ should be understoodin a distributional sense and µ∇(g ∗µ) = µg ∗∇µ as well, assuming that µ is regular enough.We may also check that this distributional definition is equivalent to defining

∇hµ(x) = P.V.

ˆRd\{x}

∇g(x− y)dµ(y).


In the case (1.5), we have

∂tFN (XtN , µ

t) = N2∂t

¨Rd×Rd

g(x− y)dµt(x)dµt(y) + ∂t∑i 6=j

g(xti − xtj)

−2N∂tN∑i=1

ˆRd

g(xti − y)dµt(y)

= −2N2ˆRd|∇hµt |2(x)dµt(x)− 2N2

ˆRd∇hµt(x) · F ∗ µt(x)dµt(x)

−2N∑i=1

∣∣∣∣∣∣∑j 6=i∇g(xti − xtj)

∣∣∣∣∣∣2

− 2N∑i=1

∑j 6=i∇g(xti − xtj) ·

∑j 6=i

F(xti − xtj)

+2N

∑i 6=j∇hµt(xti) ·

(∇g(xti − xtj) + F(xti − xtj)

)

+2NN∑i=1

P.V.

ˆRd\{xti}

(∇hµt + F ∗ µt)(x) · ∇g(x− xti)dµt(x).

We then recombine the terms to obtain

∂tFN (XtN , µ

t) = −2N2ˆRd

∣∣∣∣∣P.V.ˆRd\{x}

∇g(x− y)d(µtN − µt)(y)∣∣∣∣∣2

dµtN (x)

−2N2ˆRd|∇hµt |2dµt − 2N2

ˆRd∇hµt(x) · F ∗ µt(x)dµt(x)

−2NˆRd

(∇hµtN (x) ·

ˆRd\{x}

F(x− y)dµtN (y))dµtN (x) + 2N2

ˆRd|∇hµt |2dµtN

+2N2ˆRd∇hµt(x) ·

ˆRd\{x}

(∇g + F)(x− y)dµtN (y)dµtN (x)

+2N2ˆRdP.V.

ˆRd\{y}

(∇hµt + F ∗ µt)(x) · ∇g(x− y)dµt(x)dµtN (y).

We next recognize that all the terms except the first can be recombined and symmetrizedinto

−N2¨4c

((∇hµt + F ∗ µt)(x)− (∇hµt + F ∗ µt)(y)

)·∇g(x−y)d(µtN −µt)(x)d(µtN −µt)(y)

+ 2N2ˆRd

F ∗ (µt − µtN ) · ∇(hµtN − hµt)dµtN

which gives the desired formula.In the case (1.6) we have

∂tFN (XtN , µ

t) = N2∂t

¨g(x−y)dµt(x)dµt(y)+∂t

∑i 6=j

g(xti−xtj)−2N∂tN∑i=1

ˆRd

g(xti−y)dµt(y)

= 2N∑i 6=j∇hµt(xti) · J∇g(xti − xtj) + 2N

N∑i=1

P.V.

ˆRd\{xti}

J∇hµt(x) · ∇g(x− xti)dµt(x)


We then rewrite this as

∂tFN (XtN , µ

t) = 2N2ˆRd∇hµt(x) ·

ˆRd\{x}

J∇g(x− y)dµtN (y)dµtN (x)

+2N2ˆRdP.V.

ˆRd\{y}

J∇hµt(x) · ∇g(x− y)dµt(x)dµtN (y).

By antisymmetry of J, we recognize that the right-hand side can be symmetrized into

−N2¨4c

J(∇hµt(x)−∇hµt(y)

)· ∇g(x− y)d(µtN − µt)(x)d(µtN − µt)(y).

�

The main point is thus to control the last term in the right-hand side of (2.1) or (2.2)which is done via Proposition 1.1.

For the dissipative case with the added force, we will also need

Lemma 2.3. Assume F ∈ Hd−s

2 (Rd)∩C0,α(Rd) for some α > 0. Then there exists λ > 0 andC > 0 depending only on α, s, d such that for every t,

N2ˆRd

F∗(µt−µtN )·∇(hµtN−hµt)dµtN ≤ N2ˆRd

∣∣∣∣∣P.V.ˆRd\{x}

∇g(x− y)d(µtN − µt)(y)∣∣∣∣∣2

dµtN (x)

+C‖F‖2H

d−s2 (Rd)

(FN (Xt

N , µt) + (1 + ‖µt‖L∞)N1+ s

d + 2(N

d logN)

1(1.4)

)+C‖F‖2C0,α(Rd)N

2− 2λd .

Noting that by assumption µt ∈ ∩∞p=1Lp(Rd) and taking p to be the conjuguate exponent

to q,‖∇F ∗ µt‖L∞ ≤ ‖∇F‖Lq‖µt‖Lp ,

we then immediately deduce from Lemma 2.1, Lemma 2.3 and Proposition 1.1 that

∂tFN (XtN , µ

t) ≤ C(‖∇2hµ

t‖L∞(Rd) + ‖∇F‖Lq(Rd) + ‖F‖2H

d−s2 (Rd)

)×[ (FN (Xt

N , µt) + (1 + ‖µt‖L∞)N2− d−s

d(d+1) +N32 + 2

(N

d logN)

1(1.4)

)+C‖F‖2C0,α(Rd)N

2− 2λd

+C

(1 + ‖µt‖Cσ)N2− 1d +

(‖∇hµt‖L∞ + ‖∇2hµ

t‖L∞)‖µ‖CσN1+ s+1−σ

d]

if s ≥ d− 1

(1 + ‖µt‖L∞)N2− 1d + ‖∇hµt‖L∞‖µt‖L∞N1+ s+1

d + ‖∇2hµt‖L∞‖µt‖L∞)N1+ s

d]

if s < d− 1.

Since s < d and σ > s − d + 1, this implies by Gronwall’s lemma and in view of (1.17) thatfor every t ≤ T ,

FN (XtN , µ

t) ≤(FN (X0

N , µ0) + C1N

β)eC2t for some β < 2.

In view Proposition 3.6 below, this proves the main theorem.

3. Formulation via the electric potential

3.1. The extension representation for the fractional Laplacian. In general, the kernelg is not the convolution kernel of a local operator, but rather of a fractional Laplacian. Herewe use the extension representation popularized by [CaffSi]: by adding one space variable


y ∈ R to the space Rd, the nonlocal operator can be transformed into a local operator of theform −div (|z|γ∇·).

In what follows, k will denote the dimension extension. We will take k = 0 in the Coulombcases for which g itself is the kernel of a local operator. In all other cases, we will take k = 1.

For now, points in the space Rd will be denoted by x, and points in the extended spaceRd+k by X, with X = (x, z), x ∈ Rd, z ∈ Rk. We will often identify Rd × {0} and Rd andthus (xi, 0) with xi.

If γ is chosen such that

(3.1) d− 2 + k + γ = s,

then, given a probability measure µ on Rd, the g-potential generated by µ, defined in Rd by

hµ(x) :=ˆRd

g(x− x) dµ(x)

can be extended to a function hµ on Rd+k defined by

hµ(X) :=ˆRd

g(X − (x, 0)) dµ(x),

and this function satisfies

(3.2) − div (|z|γ∇hµ) = cd,sµδRd

where by δRd we mean the uniform measure on Rd × {0}. The corresponding values of theconstants cd,s are given in [PS, Section 1.2]. In particular, the potential g seen as a functionof Rd+k satisfies

(3.3) − div (|z|γ∇g) = cd,sδ0.

To summarize, we will take• k = 0, γ = 0 in the Coulomb cases. The reader only interested in the Coulomb cases

may thus just ignore the k and the weight |z|γ in all the integrals.• k = 1, γ = s−d + 2− k in the Riesz cases and in the one-dimensional logarithmic case

(then we mean s = 0). Note that our assumption (d − 2)+ ≤ s < d implies that γ isalways in (−1, 1). We refer to [PS, Section 1.2] for more details.

We now make a remark on the regularity of hµ:

Lemma 3.1. Assume µ is a probability density in Cθ(Rd) for some θ > s − d + 2, then wehave

(3.4) ‖∇hµ‖L∞(Rd) ≤ C(‖µ‖Cθ−1(Rd) + ‖µ‖L1(Rd)

),

and

(3.5) ‖∇2hµ‖L∞(Rd) ≤ C(‖µ‖Cθ(Rd) + ‖µ‖L1(Rd)

).

Proof. As is well known, g is (up to a constant) the kernel of ∆d−s

2 , hence hµ = cd,s∆s−d

2 µ andthe relations follow (cf. also [Du, Lemma 2.5]). �


3.2. Electring rewriting of the energy. We briefly recall the procedure used in [RS,PS]for truncating the interaction or, equivalently, spreading out the point charges. It will alsobe crucial to use the variant introduced in [LSZ, LS2] where we let the truncation distancedepend on the point.

For any η ∈ (0, 1), we define

(3.6) gη := min(g, g(η)), fη := g − gηand

(3.7) δ(η)0 := − 1

cd,sdiv (|z|γ∇gη),

which is a positive measure supported on ∂B(0, η).

Remark 3.2. This nonsmooth truncation of gη can be replaced with no change by a smoothone such that

gη(x) = g(x) for |x| ≥ η, gη(x) = cst for |x| ≤ η − ε ε <12η

and this way δ(η)0 gets replaced by a probability measure with a regular density supported in

B(0, η)\B(0, η − ε). We make this modification whenever the integrals against the singularmeasures may not be well-defined.

We will also let

(3.8) fα,η := fα − fη = gη − gα,

and we observe that fα,η has the sign of α− η, vanishes outside B(0,max(α, η)), and satisfies

(3.9) g ∗ (δ(η)x − δ(α)

x ) = fα,η(· − x)

and

(3.10) − div (|z|γ∇fα,η) = cd,s(δ(η)0 − δ(α)

0 ).

For any configuration XN = (x1, . . . , xN ), we define for any i the minimal distance

(3.11) ri = min(1

4 minj 6=i|xi − xj |, N−

1d

).

For any ~η = (η1, . . . , ηN ) ∈ RN and measure µ, we define the electric potential

(3.12) HµN [XN ] =

ˆRd+k

g(x− y)d(

N∑i=1

δxi −NµδRd

)(y)

and the truncated potential

(3.13) HµN,~η[XN ] =

ˆRd+k

g(x− y)d(

N∑i=1

δ(ηi)xi −NµδRd

)(y),

where we will quickly drop the dependence in XN . We note that

(3.14) HµN,~η[XN ] = Hµ

N [XN ]−N∑i=1

fη(x− xi).


These functions are viewed in the extended space Rd+k as described in the previous subsection,and solve

(3.15) − div (|z|γ∇HµN ) = cd,s

(N∑i=1

δxi −NµδRd

)in Rd+k,

and

(3.16) − div (|z|γ∇HµN,~η) = cd,s

(N∑i=1


)in Rd+k.

The following proposition shows how to express FN in terms of the truncated electric fields∇Hµ

N,~η. In addition, we show that the quantitiesˆRd+k|z|γ |∇Hµ

N,~η|2 − cd,s

N∑i=1

g(ηi)

converge almost monotonically (i.e. up to a small error) to FN , while the discrepancy betweenthe two can serve to control the energy of close pairs of points.

Proposition 3.3. Let µ be a bounded probability density on Rd and XN be in (Rd)N . Wemay re-write FN (XN , µ) as

(3.17) FN (XN , µ) := 1cd,s

limη→0

(ˆRd+k|z|γ |∇Hµ

N,~η|2 − cd,s

N∑i=1

g(ηi)),

and for any ~η we have the bound

(3.18)∑i 6=j

(g(xi − xj)− g(ηi))+

≤ FN (XN , µ)−(

1cd,s

ˆRd+k|z|γ |∇Hµ

N,~η|2 −

N∑i=1

g(ηi))

+ CN‖µ‖L∞N∑i=1

ηd−si ,

for some C depending only on d and s.

The proof, which is an adaptation and improvement of [PS,LS2], is postponed to Section 5.What makes our main proof work is the ability to find some choice of truncation ~η such

that´Rd+k |z|γ |∇Hµ

N,~η|2 (without the renormalizing term −cd,s

∑Ni=1 g(ηi)) is controlled by

FN (XN , µ) and the balls B(xi, ηi) are disjoint. In view of (3.18) the former could easily beachieved by taking the ηi’s large enough, say ηi = N−1/d, but the balls would not necessarilybe disjoint. Instead the choice of ηi = ri where ri are the minimal distances as in (3.11) allowsto fulfill both requirements, as seen in the following

Corollary 3.4. Under the same assumptions, we have(3.19)

N∑i=1

g(ri) ≤ C(FN (XN , µ) + (1 + ‖µ‖L∞)N1+ s

d +(N

d logN)

1(1.4)

)+ C

(N

d logN)

1(1.4)

and

(3.20)ˆRd+k|z|γ |∇Hµ

N,~r|2 ≤ C

(FN (XN , µ) + (1 + ‖µ‖L∞)N1+ s

d +(N

d logN)

1(1.4)

)for some C depending only on s, d.


Proof. Let us choose ηi = N−1/d for all i in (3.18) and observe that for each i, by definition(3.11) there exists j 6= i such that (g(|xi − xj |) − g(N−1/d))+ = (g(4ri) − g(N−1/d))+. Wemay thus write that(3.21)N∑i=1

(g(4ri)− g(N−1d ))+ ≤ FN (XN , µ)− 1

cd,s

ˆRd+k|z|γ |∇HN,~η|2 +Ng(N−

1d ) +O(N‖µ‖L∞)N

sd .

from which (3.19) follows.Let us next choose ηi = ri in (3.18). Using that ri ≤ N−1/d, this yields

0 ≤ FN (XN , µ)− 1cd,s

ˆRd+k|z|γ |∇HN,~r|2 +

N∑i=1

g(ri) +O(N‖µ‖L∞)Nsd .

Combining with (3.19), (3.20) follows.�

From (3.20) we directly obtain that FN is bounded below:

Corollary 3.5. Under the same assumptions we have

(3.22) FN (XN , µ) ≥ −(N

d logN)

1(1.4) − CN1+ sd

for some C > 0 depending only on d, s and ‖µ‖L∞ .

3.3. Coerciveness of the modulated energy. Here we prove that the modulated energydoes metrize the convergence of µtN to µt.

Proposition 3.6. For any 0 < α ≤ 1, there exists λ > 0 and C > 0 depending only on α, d,s, such that for any XN ∈ (Rd)N , any probability density µ, and any ξ ∈ C∞(Rd), we have

(3.23)∣∣∣∣∣ˆRdξd

(N∑i=1

δxi −Nµ)∣∣∣∣∣ ≤ C‖ξ‖C0,α(Rd)N

1−λd

+ C‖ξ‖H

d−s2 (Rd)

(FN (XN , µ) + (1 + ‖µ‖L∞)N1+ s

d + 2(N

d logN)

1(1.4)

) 12.

In particular, if 1N2FN (XN , µ)→ 0 as N →∞, we have that

(3.24) 1N

N∑i=1

δxi ⇀ µ in the weak sense.

Proof. Let ξ be a smooth test function on Rd. Let ξ denote an extension of ξ to Rd+k satisfying

−div (|z|γ∇ξ) = 0 in {z 6= 0}.

By [FKS], |z|γ being a Muckenhoupt A2 weight, the function ξ is in C0,λ(Rd+k) for someλ > 0 depending on the other parameters, with ‖ξ‖C0,λ ≤ C‖ξ‖C0,α . This can also be seenfrom the Poisson kernel representation given in [CaffSi]. In addition, we also have (this canbe seen in Fourier, see [CaffSi, Section 3.2]

(3.25)ˆRd+k|z|γ |∇ξ|2 = C‖ξ‖2

Hd−s

2 (Rd).


Using (3.16) let us write for any probability density µ

(3.26)ˆRdξ d

(N∑i=1

δxi −Nµ)

=ˆRdξd

(N∑i=1

δxi − δ(ri)xi

)− 1

cd,s

ˆRd+k

ξ div (|z|γ∇HµN,~r[XN ]).

For the first term in the right-hand side we use the Holder continuity of ξ and the fact thatδ

(ri)xi is supported in B(xi, ri) and write

(3.27)∣∣∣∣∣ˆRdξd

N∑i=1

(δxi − δ(ri)

xi

)∣∣∣∣∣ ≤ C‖ξ‖C0,α(Rd)

N∑i=1

rλi ≤ C‖ξ‖C0,α(Rd)N1−λd .

For the second term, we integrate by parts and use the Cauchy-Schwarz inequality to write

(3.28)∣∣∣∣ˆ

Rd+kξ div (|z|γ∇Hµ

N,~r[XN ])∣∣∣∣ =

∣∣∣∣ˆRd+k|z|γ∇ξ · ∇Hµ

N,~r

∣∣∣∣≤(ˆ

Rd+k|z|γ |∇ξ|2

) 12(ˆ

Rd+k|z|γ |∇Hµ

N,~r|2) 1

2

In view of (3.20) and (3.25) we thus find

(3.29)∣∣∣∣ˆ

Rd+kξ div (|z|γ∇Hµ

N,~r[XN ])∣∣∣∣

≤ C‖ξ‖H

d−s2 (Rd)

(FN (XN , µ) + (1 + ‖µ‖L∞)N1+ s

d + 2(N

d logN)

1(1.4)

) 12.

Inserting (3.27) and (3.29) into (3.26) we conclude the result. �

Remark 3.7. In a density formulation aiming at proving propagation of chaos, arguingexactly as in [RS, Lemma 8.4] for instance, we may deduce from this result and the maintheorem the convergence of the k-marginal densities in the dual of some Sobolev space, withrate k/N times the right-hand side of (3.23).

3.4. Proof of Lemma 2.3. Using the Cauchy-Schwarz inequality we have the bound¨4c

F ∗ (µt − µtN ) · ∇g(x− y)d(µtN − µt)(y)dµtN (x)

≤(ˆ

Rd|F ∗ (µt − µtN )|2dµtN

) 12

ˆ ∣∣∣∣∣P.V.ˆRd\{x}

∇g(x− y)d(µtN − µt)(y)∣∣∣∣∣2

dµtN (x)

12

thus to prove the lemma, it suffices to show that

(3.30) N‖F ∗ (µt − µtN )‖L∞ ≤ C‖F‖C0,α(Rd)N1−λd

+ C‖F‖H

d−s2 (Rd)

(FN (XN , µ) + (1 + ‖µ‖L∞)N1+ s

d + 2(N

d logN)

1(1.4)

) 12,

which is a direct consequence of (3.23).


4. Proof of Proposition 1.1

4.1. Stress-energy tensor.

Definition 4.1. For any functions h, f in Rd+k such that´Rd+k |z|γ |∇h|2 and

´Rd+k |z|γ |∇f |2

are finite, we define the stress tensor [h, f ] as the (d + k)× (d + k) tensor(4.1) [h, f ] = |z|γ (∂ih∂jf + ∂ih∂jf)− |z|γ∇h · ∇fδijwhere δij = 1 if i = j and 0 otherwise.

We note that

Lemma 4.2. If h and f are regular enough, we have(4.2) div [h, f ] = div (|z|γ∇h)∇f + div (|z|γ∇f)∇h−∇|z|γ∇h · ∇fwhere divT here denotes the vector with components

∑i ∂iTij, with j ranging from 1 to d + k.

Proof. This is a direct computation. Below, all sums range from 1 to d + k.∑i

∂i[h, f ]ij

=∑i

[∂i(|z|γ∂ih)∂jf + ∂i(|z|γ∂if)∂jh+ |z|γ∂ijh∂if + |z|γ∂ijf∂ih]− ∂j

(|z|γ

∑i

∂ih∂if

)= div (|z|γ∇h) + div (|z|γ∇f)−∇h · ∇f∂j |z|γ .

�

In view of (4.2), we have

Lemma 4.3. Let ψ : Rd → Rd be Lipschitz, and if k = 1 let ψ be an extension of it to a mapfrom Rd+k to Rd+k, whose last component identically vanishes, which tends to 0 as |z| → ∞and has the same pointwise and Lipschitz bounds as ψ. 2 For any measures µ, ν on Rd+k, if−div (|z|γ∇hµ) = cd,sµ and −div (|z|γ∇hν) = cd,sν, and assuming that

´Rd+k |z|γ |∇hµ|2 and´

Rd+k |z|γ |∇hν |2 are finite and the left-hand side in (4.3) is well-defined, we have

(4.3)¨

Rd+k×Rd+k(ψ(x)− ψ(y)) · ∇g(x− y)dµ(x)dν(y) = 1

cd,s

ˆRd+k∇ψ(x) : [hµ, hν ].

Proof. If µ is smooth enough then we may use (4.2) to write¨Rd+k×Rd+k

(ψ(x)− ψ(y)) · ∇g(x− y)dµ(x)dν(y) =ˆRd+k

ψ · (∇hµdν +∇hνdµ)

= − 1cd,s

ˆRd+k

ψ · div [hµ, hν ]

since the last component of ψ vanishes identically. Integrating by parts, we obtain¨Rd+k×Rd+k

(ψ(x)− ψ(y)) · ∇g(x− y)dµ(x)dν(y) = 1cd,s

ˆRd+k∇ψ : [hµ, hν ].

By density, we may extend this relation to all measures µ, ν such that both sides of (4.3)make sense. �

2Such an extension exists, for instance by solving the∞-Laplacian in a strip, which provides an “absolutelyminimal Lipschitz extension”


4.2. Proof of Proposition 1.1. We now proceed to the proof. Given the Lipschitz mapψ : Rd → Rd, we choose an extension ψ to Rd+k which satisfies the same conditions as inLemma 4.3.

Step 1: renormalizing the quantity and expressing it with the stress-energy tensor. Clearly,

(4.4)¨4c

(ψ(x)− ψ(y)) · ∇g(x− y)d(N∑i=1

δxi −Nµ)(x)d(N∑i=1

δxi −Nµ)(y)

= limη→0

[¨Rd+k×Rd+k

(ψ(x)− ψ(y)

)· ∇g(x− y)d(

N∑i=1

δ(η)xi −NµδRd)(x)d(

N∑i=1

δ(η)xi −NµδRd)(y)

−N∑i=1

¨Rd+k×Rd+k

(ψ(x)− ψ(y)

)· ∇g(x− y)dδ(η)

xi (x)dδ(η)xi (y)

].

Applying Lemma 4.3, in view of (3.13), (3.16) we find that

(4.5)¨

Rd+k×Rd+k

(ψ(x)− ψ(y)

)· ∇g(x− y)d(

N∑i=1

δ(η)xi −NµδRd)(x)d(

N∑i=1

δ(η)xi −NµδRd)(y)

= 1cd,s

ˆRd+k∇ψ : [Hµ

N,~η, HµN,~η].

Step 2: analysis of the diagonal terms. The main point is to understand how they varywith ηi. Let ~α be such that αi ≥ ηi for every i.

We may write that

(4.6)¨

Rd+k×Rd+k

(ψ(x)− ψ(y)

)· ∇g(x− y)dδ(ηi)

xi (x)dδ(ηi)xi (y)

−¨

Rd+k×Rd+k

(ψ(x)− ψ(y)

)· ∇g(x− y)dδ(αi)

xi (x)dδ(αi)xi (y)

=¨

Rd+k×Rd+k(ψ(x)− ψ(y)) · ∇g(x− y)d(δ(ηi)

xi − δ(αi)xi )(x)d(δ(ηi)

xi − δ(αi)xi )(y)

+ 2¨

Rd+k×Rd+k(ψ(x)− ψ(y)) · ∇g(x− y)dδ(αi)

xi (x)d(δ(ηi)xi − δ

(αi)xi )(y).

We claim that

(4.7)¨

Rd+k×Rd+k(ψ(x)− ψ(y)) · ∇g(x− y)dδ(αi)


(αi)xi )(y) = 0.

Assuming this, inserting it to (4.6) and using (3.9) and Lemma 4.3, we conclude that

(4.8)¨

Rd+k×Rd+k

(ψ(x)− ψ(y)

)· ∇g(x− y)dδ(η)

xi (x)dδ(η)xi (y)

−¨

Rd+k×Rd+k

(ψ(x)− ψ(y)


xi (x)dδ(αi)xi (y)

= 1cd,s

ˆRd+k∇ψ : [fαi,ηi(· − xi), fαi,ηi(· − xi)].


Step 3: proof of (4.7). Let us write the quantity in (4.7) as

(4.9) 2¨

Rd+k×Rd+k(ψ(x)− ψ(y)) · ∇g(x− y)dδ(2αi)


(αi)xi )(y)

+ 2¨

Rd+k×Rd+k(ψ(x)− ψ(y)) · ∇g(x− y)d

(δ(αi)xi − δ

(2αi)xi

)(x)d

(δ(ηi)xi − δ

(αi)xi

)(y).

In view of (3.7), ∇g ∗ δ(2αi)xi = ∇g2αi(· − xi) and in view of (3.9), ∇g ∗ (δ(ηi)

xi − δ(αi)xi ) =

∇fαi,ηi(x− xi). We may thus rewrite the first term in (4.9) as

2P.V.ˆRd+k

ψ · ∇fαi,ηi(· − xi)dδ(2αi)xi + 2P.V.

ˆRd+k

ψ · ∇g2αi(· − xi)d(δ(ηi)xi − δ

(αi)xi

).

But ∇fαi,ηi(· − xi) is supported in B(xi, αi) while δ(2αi)xi is supported on ∂B(xi, 2αi), and

in the same way ∇gαi(· − xi) vanishes in B(xi, 2αi) where δ(ηi)xi − δ

(αi)xi is supported, so we

conclude that the first term in (4.9) is zero. The second term in (4.9) is equal by (3.9) andLemma 4.3 to

1cd,s

ˆRd+k∇ψ : [f2αi,αi(· − xi), fαi,ηi(· − xi)]

and it is zero, since f2αi,αi and fαi,ηi have disjoint supports. This finishes the proof of (4.7).Step 4: combining (4.5) and (4.8). The following lemma allows to recombine the terms

obtained at different values of ηi while making only a small error.

Lemma 4.4. Assume that µ ∈ Cσ(Rd) with σ > s− d + 1 if s ≥ d− 1. Assume µ ∈ L∞(Rd)or µ ∈ Cσ(Rd) with σ > 0 if s < d− 1. If for each i we have ηi < αi ≤ ri, thenˆRd+k∇ψ : [Hµ

N,~η, HµN,~η] =

ˆRd+k∇ψ : [Hµ

N,~α, HµN,~α]+

N∑i=1

ˆRd+k∇ψ : [fαi,ηi(·−xi), fαi,ηi(·−xi)]+E

with

(4.10) |E| ≤ C‖∇ψ‖L∞(FN (XN , µ) +

(N

d logN)

1(1.4) + (1 + ‖µ‖L∞)N2− d−sd(d+1)

)

+CN min(‖ψ‖L∞‖µ‖L∞

N∑i=1

αd−s−1i + ‖∇ψ‖L∞‖µ‖L∞

N∑i=1

αd−si , ‖ψ‖W 1,∞‖µ‖Cσ

N∑i=1

αd−s+σ−1i

)

+ CN

{‖∇ψ‖L∞(1 + ‖µ‖Cσ)

∑Ni=1 αi if s ≥ d− 1

‖∇ψ‖L∞(1 + ‖µ‖L∞)∑Ni=1 αi if s < d− 1

where C depends only on s, d.

Assuming this, and combining (4.4), (4.5) and (4.8) we find that for any αi ≤ ri,¨4c

(ψ(x)− ψ(y))·∇g(x−y)d(N∑i=1

δxi−Nµ)(x)d(N∑i=1

δxi−Nµ)(y) = 1cd,s

ˆRd+k∇ψ : [Hµ

N,~α, HµN,~α]

−N∑i=1

¨Rd+k×Rd+k

(ψ(x)− ψ(y)


xi (x)dδ(αi)xi (y) +O(E)


where E is as in (4.10). Using the Lipschitz character of ψ and the expression of g, we findthat the second term on the right-hand side can be bounded by 3

C‖∇ψ‖L∞N∑i=1

¨Rd+k×Rd+k

g(x− y)dδ(αi)xi (x)dδ(αi)

xi (y)

= C‖∇ψ‖L∞N∑i=1

ˆRd+k

gαi(· − xi)dδ(αi)xi = C‖∇ψ‖L∞

N∑i=1

g(αi)

where we have used (3.7). Choosing finally αi = ri ≤ N−1/d, bounding pointwise [HµN,~r, H

µN,~r]

by 2|z|γ |∇HµN,~r|

2 and using (3.20), while using (3.19) to bound∑Ni=1 g(ri), we conclude the

proof of Proposition 1.1.

Proof of Lemma 4.4. First, we observe from (3.14) that [HµN,~α, H

µN,~α] and [Hµ

N,~η, HµN,~η] only

differ in the balls B(xi, αi) which are disjoint since αi ≤ ri, and that in each B(xi, αi) we haveHµN,~η = Hµ

N,~α + fαi,ηi(· − xi).

We thus deduce that

(4.11)ˆB(xi,αi)

∇ψ :([Hµ

N,~η, HµN,~η]− [Hµ

N,~α, HµN,~α]

)=ˆB(xi,αi)

∇ψ :([fαi,ηi , fαi,ηi ](· − xi) + 2[fαi,ηi(· − xi), H

µN,~α]

)=ˆRd+k∇ψ :

([fαi,ηi , fαi,ηi ](· − xi) + 2[fαi,ηi(· − xi), H

µN,~α]

).

There only remains to control the second part of the right-hand side. By Lemma 4.3, we haveˆRd+k∇ψ : [fαi,ηi(· − xi), H

µN,~α]

= cd,s

¨Rd+k×Rd+k

(ψ(x)− ψ(y)) · ∇g(x− y)d

N∑j=1

δ(αj)xj −NµδRd

(x)d(δ(ηi)xi − δ

(αi)xi

)(y).

In view of (4.7), we just need to bound the sum over i of

(4.12)

cd,s

¨Rd+k×Rd+k

(ψ(x)− ψ(y)) · ∇g(x− y)d

∑j:j 6=i


(x)d(δ(ηi)xi − δ

(αi)xi

)(y)

= cd,s

ˆRd+k

ψ · ∇fαi,ηi(· − xi)d

∑j:j 6=i


+ cd,s

ˆRd+k

ψ ·

∑j:j 6=i∇gαj (x− xj)−N∇hµ

d (δ(ηi)xi − δ

(αi)xi

),

where we used (3.9).

3In the case (1.4) we bound instead |x− y||∇g(x− y)| by 1, which yields an even better control.


Step 1: first term in (4.12). Since fαi,ηi(·−xi) is supported in B(xi, αi), δ(αj)xj in B(xj , αj)

and the balls are disjoint, one type of terms vanishes and there remains

−Ncd,s

ˆRdψ · ∇fαi,ηi(· − xi)dµ.

Thanks to the explicit form of fα,η we have

fαi,ηi(· − xi) ={

g(x− xi)− g(αi) for ηi < |x− xi| ≤ αig(ηi)− g(αi) for |x− xi| ≤ ηi

and∇fαi,ηi(· − xi) = ∇g(x− xi)1ηi≤|x−xi|≤αi .

It follows that

(4.13)ˆRd|fα| ≤ Cαd−s,

ˆRd|fαi,ηi | ≤ Cαd−s

i ,

ˆRd|∇fαi,ηi | ≤ Cαd−s−1

i .

Indeed, it suffices to observe that

(4.14)ˆB(0,η)

fα = C

ˆ α

0(g(r)− g(α))rd−1 dr = −Cd

ˆ r

0g′(r)rd dr,

with an integration by parts.We may always write

(4.15)∣∣∣∣ˆ

Rdψ · ∇fαi,ηi(· − xi)(µ− µ(xi))

∣∣∣∣ ≤ C‖ψ‖L∞‖µ‖C0,1

ˆ αi

ηi

rd

rs+1dr

≤ C‖ψ‖L∞‖µ‖C0,1αd−si ,

and, integrating by parts and using (4.13),

(4.16)∣∣∣∣ˆ

Rdψ · ∇fαi,ηi(· − xi)µ(xi)

∣∣∣∣ ≤ ‖µ‖L∞‖∇ψ‖L∞ ˆRd|fαi,ηi | ≤ ‖µ‖L∞‖∇ψ‖L∞αd−s

i .

Alternatively, we may use the simpler bound derived from (4.13),

(4.17)∣∣∣∣ˆ

Rdψ · ∇fαi,ηi(· − xi)dµ

∣∣∣∣ ≤ C‖ψ‖L∞‖µ‖L∞αd−s−1i .

A standard interpolation argument yields that ‖g‖(Cσ)∗ ≤ ‖g‖σ(C1)∗‖g‖1−σ(C0)∗ so interpolating

between (4.15)–(4.16) and (4.17), we obtain∣∣∣∣ˆRdψ · ∇fαi,ηi(· − xi)dµ

∣∣∣∣ ≤ C‖ψ‖1−σL∞ ‖ψ‖σW 1,∞‖µ‖Cσαd−s+σ−1

i .

We conclude that the sum over i of the first terms in (4.12) is bounded by both

(4.18) C‖ψ‖1−σL∞ ‖ψ‖σW 1,∞‖µ‖Cσ

∑i

αd−s+σ−1i and C‖ψ‖L∞‖µ‖L∞

∑i

αd−s−1i .


Step 2: second term in (4.12). We may rewrite the integral as

(4.19) −NˆRd+k

ψ(xi) · ∇Rdhµd(δ(ηi)xi − δ

(αi)xi

)+∑j:j 6=i

ˆRd+k

ψ(xj) · ∇gαj (x− xj)d(δ(ηi)xi − δ

(αi)xi

)+∑j:j 6=i

ˆRd+k

(ψ − ψ(xj)) · ∇gαj (x− xj)d(δ(ηi)xi − δ

(αi)xi

)+O

(N‖∇ψ‖L∞

ˆRd+k|x− xi| |∇Rdhµ| d

(δ(ηi)xi + δ(αi)

xi

)),

where we used that the last component of ψ vanishes, so that only the derivatives along theRd directions appear.

Substep 2.1: first term of (4.19). We may write that δ(ηi)xi −δ

(αi)xi = − 1

cd,sdiv (|z|γ∇fαi,ηi(·−

xi)) and integrate by parts twice to get1

cd,s

ˆRd+k|z|γ∇

(ψ(xi) · ∇Rdhµ

)· ∇fαi,ηi(· − xi) =

ˆRd

(ψ(xi) · ∇µ)fαi,ηi .

Here, we used that −div (|z|γ∇hµ) = cd,sµδRd and took the ψ(xi) · ∇Rd of this relation. Inview of (4.13), this is then bounded by

C‖ψ‖L∞‖∇µ‖L∞αd−si .

Alternatively, we may integrate by parts in Rd to bound it by

C‖µ‖L∞(‖ψ‖L∞

ˆRd|∇fαi,ηi |+ ‖∇ψ‖L∞αd−s

i

).

Interpolating as above, we conclude with (4.13) that the sum over i of these terms is boundedby both

C‖µ‖Cσ(‖ψ‖σL∞‖∇ψ‖1−σL∞ α

d−si + ‖ψ‖L∞

∑i

αd−s+σ−1i

)(4.20)

and ‖µ‖L∞(‖ψ‖L∞

∑i

αd−s−1i + ‖∇ψ‖L∞

∑i

αd−si

).

Substep 2.2: second term of (4.19). Arguing in the same way as for the first term, usingthat −div (|z|γ∇gαj (· − xj)) = cd,sδ

(αj)xj and the disjointness of the balls, we find that this

term vanishes.Substep 2.3: third term in (4.19). We separate the sum into two pieces and bound this

term by

(4.21)∑

j 6=i,|xi−xj |≥N−εd

ˆRd+k

(ψ − ψ(xj)) · ∇gαj (· − xj)d(δ(ηi)xi − δ

(αi)xi

)

+ ‖∇ψ‖L∞∑

j 6=i,|xi−xj |≤N−εd

ˆRd+k|x− xj ||∇gαj (x− xj)|d

(δ(ηi)xi + δ(αi)

xi

)(x),


for some ε > 0 to be determined. For the first term of (4.21), we may use that |xi−xj | ≥ N−εd

to write‖∇

((ψ − ψ(xj)) · ∇gαj (x− xj)

)‖L∞(B(xi,ri)) ≤ C‖∇ψ‖L∞N

ε(s+1)d

and using that ηi ≤ αi ≤ N−1d we may thus bound the sum of such terms by

C‖∇ψ‖L∞Nε(s+1)

d +2− 1d .

Since |∇gα| ≤ |∇g|, we may bound the second term in (4.21) by

(4.22) ‖∇ψ‖L∞∑

j 6=i,|xi−xj |≤N−εd

|xi − xj |−s.

To bound this, let us choose ηi = 2N−εd in (3.18) to obtain that∑

i 6=j

(g(xi − xj)− g(2N−

εd ))

+≤ FN (XN , µ) +Ng(2N−

εd ) + C‖µ‖L∞N2− ε(d−s)

d .

In the cases (1.3), it follows that∑i 6=j,|xi−xj |≤N−

εd

g(xi − xj) ≤ C(FN (XN , µ) +N1+ (d−s)ε

d + ‖µ‖L∞N2− ε(d−s)d

).

In the cases (1.4), it follows that∑i 6=j,|xi−xj |≤N−

εd

log 2 ≤ FN (XN , µ) + N

d logN + C(1 + ‖µ‖L∞)N,

and this suffices to bound (4.22) as well. Choosing ε = 1d+1 , we conclude in all cases that the

sum over i of the third terms in (4.19) is bounded by

(4.23) C‖∇ψ‖L∞(FN (XN , µ) + (1 + ‖µ‖L∞)N2− d−s

d(d+1) +(N

d logN)

1(1.4)

).

Substep 2.4: fourth term in (4.19). We may bound it byO(‖∇ψ‖L∞αi‖∇Rdhµ‖L∞(Rd+k)

).

But since hµ = g ∗ µ, it is straightforward to check that ‖∇Rdhµ‖L∞(Rd+k) ≤ ‖∇hµ‖L∞(Rd).Using (3.4), we conclude the sum of these terms is bounded by

(4.24){C∑i αi‖∇ψ‖L∞(1 + ‖µ‖L∞) if s < d− 1

C∑i αi‖∇ψ‖L∞(1 + ‖µ‖Cσ) if s ≥ d− 1 and σ > s− d + 1.

Combining the bounds (4.18), (4.20), (4.23), (4.24) concludes the proof of the lemma. �

5. Proof of Proposition 3.3

We drop the superscripts µ. First,´Rd+k |z|γ |∇HN,~η|2 is a convergent integral and

(5.1)ˆRd+k|z|γ |∇HN,~η|2 = cd,s

¨Rd+k×Rd+k

g(x−y)d(

N∑i=1


)(x)d

(N∑i=1


)(y).


Indeed, we may choose R large enough so that all the points of XN are contained in the ballBR = B(0, R) in Rd+k. By Green’s formula and (3.16), we have

ˆBR

|z|γ |∇HN,~η|2 =ˆ∂BR

|z|γHN,~η∂HN

∂n+ cd,s

ˆBR

HN,~η d

(N∑i=1


).

Since´d(∑i δxi−Nµ) = 0, the function HN,~η decreases like 1/|x|s+1 and ∇HN,~η like 1/|x|s+2

as |x| → ∞, hence the boundary integral tends to 0 as R→∞, and we may write

ˆRd+k|z|γ |∇HN,~η|2 = cd,s

ˆRd+k

HN,~η d

(N∑i=1


)

and thus by (3.16), (5.1) holds. We may next write that

limη→0

[¨Rd+k×Rd+k

g(x− y)d(

N∑i=1


)(x)d

(N∑i=1


)(y)−

N∑i=1

g(ηi)]

=¨4c

g(x− y)d(

N∑i=1

δxi −NµδRd

)(x)d

(N∑i=1

δxi −NµδRd

)(y)

and we deduce in view of (5.1) that (3.17) holds.We next turn to the proof of (3.18), adapted from [PS]. From (3.16) applied with ~η = ~α

and in view of (3.13), we have ∇HN,~η = ∇HN,~α +∑Ni=1∇fαi,ηi(· − xi) thus

ˆRd+k|z|γ |∇HN,~η|2 =

ˆRd+k|z|γ |∇HN,~α|2 +

∑i,j

ˆRd+k|z|γ∇fαi,ηi(x− xi) · ∇fαi,ηi(x− xj)

+ 2N∑i=1

ˆRd+k|z|γ∇fαi,ηi(x− xi) · ∇HN,~α.

Using (3.10), we first write

∑i,j

ˆRd+k|z|γ∇fαi,ηi(x− xi) · ∇fαi,ηi(x− xj)

= −∑i,j

ˆRd+k

fαi,ηi(x− xi)div (|z|γ∇fαi,ηi(x− xj)) = cd,s∑i,j

ˆRd+k

fαi,ηi(x− xi)d(δ(ηi)xj − δ

(αi)xj ).

Next, using (3.16), we write

2N∑i=1

ˆRd+k|z|γ∇fαi,ηi(x− xi) · ∇HN,~α = −2

N∑i=1

ˆRd+k

fαi,ηi(x− xi)div (|z|γ∇HN,~α)

= 2cd,s

N∑i=1

ˆRd+k

fαi,ηi(x− xi) d( N∑j=1

δ(αi)xj − µδRd

).


These last two equations add up to give a right-hand side equal to

(5.2)∑i 6=j

cd,s

ˆRd+k

fαi,ηi(x− xi)d(δ(αi)xj + δ

(ηj)xj )− 2cd,s

N∑i=1

ˆfαi,ηi(x− xi)dµδRd

+Ncd,s

ˆRd+k

fαi,ηid(δ(αi)0 + δ

(ηi)0 ) .

We then note that´

fαi,ηi d(δ(αi)0 + δ

(ηi)0 ) = −

´fηidδ

(αi)0 = −(g(αi)− g(ηi)) by definition of fη

and the fact that δ(α)0 is a measure supported on ∂B(0, α) and of mass 1. Secondly, by (4.13)

we may bound´Rd fαi,ηi(x− xi)µδRd by C‖µ‖L∞αd−s

i .Thirdly, we observe that since fαi,ηi ≤ 0, the first term in (5.2) is nonpositive and we may

bound it above by

∑i 6=j

cd,s

ˆRd+k

fαi,ηidδ(αj)xj ≤

∑i 6=j

cd,s

ˆRd+k

(gηi(x− xi)− gαi(x− xi)) dδ(αj)xj

≤∑i 6=j

cd,s

ˆRd+k

(g(ηi)− gαi(|xi − xj |+ αj))−

where we used the fact that gα is radial decreasing. Combining the previous relations yields

− CN‖µ‖L∞N∑i=1

ηd−si + cd,s

∑i 6=j

(gαi(|xi − xj |+ αj)− g(ηi))+

≤(ˆ

Rd+k|z|γ |∇HN,~α|2 − cd,s

N∑i=1

g(αi))−(ˆ

Rd+k|z|γ |∇HN,~η|2 − cd,s

N∑i=1

g(ηi))

and letting all αi → 0 finishes the proof in view of (3.17).

Appendix A. Mean-field limit for monokinetic Vlasov systems,with Mitia Duerinckx

In this appendix, we turn to examining the mean-field limit of solutions of Newton’s second-order system of ODEs, that is,

xi = vi,

vi = − 1N∇xiHN (x1, . . . , xN ),

xi(0) = x0i , vi(0) = v0

i ,

i = 1, . . . , N(A.1)

where HN is the interaction energy defined in (1.3). We then consider the phase-space em-pirical measure

f tN := 1N

N∑i=1

δ(xti,vti) ∈ P(Rd × Rd),

where P denotes probability measures, associated to a solution ZtN := ((xt1, vt1), . . . , (xtN , vtN ))of the flow (A.1). If the points (x0

i , v0i ), which themselves depend on N , are such that f0

N

converges to some regular measure f0, then formal computations indicate that for t > 0, f tN


should converge to the solution of the Cauchy problem with initial data f0 for the followingVlasov equation,

∂tf + v · ∇xf + (∇g ∗ µ) · ∇vf = 0, µt(x) :=ˆRdf t(x, v) dv.(A.2)

In the case of a smooth interaction kernel g, such a mean-field result was first establishedin the 1970s by a weak compactness argument [NW, BH], while Dobrushin [Do] gave thefirst quantitative proof in 1-Wasserstein distance. In recent years much attention has beengiven to the physically more relevant case of singular Coulomb-like kernels, but only verypartial results have been obtained. In [HJ2, HJ1], exploiting a Gronwall argument on the∞-Wasserstein distance between fN and f , Hauray and Jabin treated all interaction kernelsg satisfying |∇g(x)| ≤ C|x|−s and |∇2g(x)| ≤ C|x|−s−1 for some s < 1. In [JW1], Jabinand Wang introduced a new approach, allowing them to treat all interaction kernels withbounded gradient. The same problem has been addressed in [BP, La, LP], leading to resultsthat require a small N -dependent cutoff of the interaction kernel.

In this appendix, we show that the method presented in the article allows to unlock themean-field problem with Coulomb-like interaction in the simpler case of monokinetic data,that is, if there exists a regular velocity field u0 : Rd → Rd such that v0

i ≈ u0(x0i ) for

i = 1, . . . , N , which implies that the solutions remain monokinetic, at least for short times.Justifying a complete mean-field result for non-monokinetic solutions in the Coulomb casein dimensions d ≥ 2 remains one of the main open problems in the field. It is expected tobe rendered difficult by the possibility of concentration and filamentation in both space andvelocity variables.

In the monokinetic setting, we focus on the (spatial) empirical measure

µtN := 1N

N∑i=1

δxti∈ P(Rd),

associated to a solution ZtN of the flow (A.1). If the points x0i are such that µ0

N converges tosome regular measure µ0, then formal computations (see for instance [LZ]) lead to expectingthat for t > 0, µtN converges to the solution µt of the Cauchy problem with initial data (µ0, u0)of the following monokinetic version of (A.2)4,

∂tµ+ div (µu) = 0, ∂tu+ u · ∇u = −∇g ∗ µ.(A.3)

In the Coulomb case, these equations are known as the pressureless (repulsive) Euler-Poissonsystem. Since for the second-order system (A.1) the total energy splits into a potential and akinetic part, we naturally introduce a modulated total energy taking both parts into account:for ZN := ((x1, v1), . . . , (xN , vN )) and for a couple (µ, u) ∈ P(Rd)× C(Rd), we set

HN (ZN , (µ, u)) := NN∑i=1|u(xi)− vi|2

+¨

Rd×Rd\4g(x− y) d

( N∑i=1

δxi −Nµ)(x) d

( N∑i=1

δxi −Nµ)(y),

4Formally, the pair (µ, u) indeed satisfies the system (A.3) if and only if the monokinetic ansatz f t(x, v) :=µt(x)δv=ut(x) satisfies the Vlasov equation (A.2).


and we view HN (ZtN , (µt, ut)) as a good notion of distance between µtN and µt that is adaptedto the monokinetic setting. The condition HN (ZtN , (µt, ut)) = o(N2) indeed entails the weakconvergence µtN ⇀ µt as in Proposition 3.6, while due to the kinetic part it also implies thatthe flow ZtN remains monokinetic, that is, vti ≈ ut(xti). In parallel with Theorem 1, our mainresult is a Gronwall inequality on the time-derivative of HN (ZtN , (µt, ut)), which implies aquantitative rate of convergence of µtN to µt in that metric.

Theorem 2. Assume that g is of the form (1.3) or (1.4). Assume that (A.3) admits asolution (µ, u) in [0, T ] × Rd for some T > 0, with µ ∈ L∞([0, T ];P ∩ L∞(Rd)) and u ∈L∞([0, T ];W 1,∞(Rd)d). In the case s ≥ d − 1, also assume µ ∈ L∞([0, T ];Cσ(Rd)) for someσ > s − d + 1. Let ZtN solve (A.1). Then there exist constants C1, C2 depending only oncontrolled norms of (µ, u) and on d, s, σ, and there exists an exponent β < 2 depending onlyon d, s, σ, such that for every t ∈ [0, T ] we have

HN (ZtN , (µt, ut)) ≤(HN (Z0

N , (µ0, u0)) + C1Nβ)eC2t.

In particular, if µ0N ⇀ µ0 and is such that

limN→∞

1N2HN (Z0

N , µ0) = 0,

then the same is true for every t ∈ [0, T ], hence µtN ⇀ µt in the weak sense.

Remark A.1. We briefly comment on the regularity assumptions on the solution (µ, u) ofthe mean-field system (A.3) in the Coulomb case, that is, the pressureless (repulsive) Euler-Poisson system. The local-in-time existence of a smooth solution is easily established bystandard Besov techniques, e.g. [BCD, Section 3.2]: for non-integer σ > 0, given initial dataµ0 ∈ P∩Cσ(Rd) and u0 ∈ Cσ+1(Rd), there exists some T > 0 such that (A.3) admits a uniquesolution (µ, u) in [0, T ]× Rd with µ ∈ L∞([0, T ];P ∩ Cσ(Rd)) and u ∈ L∞([0, T ];Cσ+1(Rd)).While global existence of a weak solution was proved in [NT], together with a partial uniquenessresult, the solution can in general remain smooth only locally in time due to possible wavebreakdown [Pe, En]. Global smoothness was however established in some cases [CW, Gu].The precise configuration of initial data happens to play a decisive role: a critical thresholdphenomenon was discovered by Engelberg, Liu, and Tadmor [ELT], showing in dimension 1that global smoothness holds whenever the initial data do not cross some intrinsic criticalthreshold, while finite-time breakdown occurs otherwise. In particular, global smoothness isexpected to hold for a large set of initial configurations, but only partial results in this veinare still available in dimension d > 1, cf. [En, LT1, LT2, LL].

Before turning to the proof of Theorem 2, we start with a weak-strong stability principlefor the mean-field system (A.3), analogous to (1.21) for the mean-field equation (1.13) in thefirst-order case.

Lemma A.2. Let (µt1, ut1) and (µt2, ut2) denote two smooth solutions of (A.3) in [0, T ]× Rd.Then there is a constant C depending only on d such that, in terms of

H((µt1, ut1), (µt2, ut2)) :=ˆRd|ut1 − ut2|2µt1 +

ˆRd

ˆRd

g(x− y) d(µt1 − µt2)(x) d(µt1 − µt2)(y),

we haveH((µt1, ut1), (µt2, ut2)) ≤ eC

´ t0 ‖∇u

u2‖L∞duH((µ0

1, u01), (µ0

2, u02)).


Proof. Let hi := g ∗ µi. We compute

∂tH((µ1, u1), (µ2, u2)) =ˆRd|u1 − u2|2∂tµ1 + 2

ˆRd

(u1 − u2) · (∂tu1 − ∂tu2)µ1

+2ˆRd

(h1 − h2)(∂tµ1 − ∂tµ2)

=ˆRd∇|u1 − u2|2 · µ1u1 − 2

ˆRd

(u1 − u2) · (u1 · ∇u1 − u2 · ∇u2)µ1

−2ˆRd∇(h1 − h2) · (u1 − u2)µ1 + 2

ˆRd∇(h1 − h2) · (µ1u1 − µ2u2).

Decomposing 2(u1−u2) · (u1 ·∇u1−u2 ·∇u2) = u1 ·∇|u1−u2|2 +2∇u2 : (u1−u2)⊗ (u1−u2),we find after straightforward simplifications,

∂tH((µ1, u1), (µ2, u2))

= −2ˆRdµ1∇u2 : (u1 − u2)⊗ (u1 − u2) + 2

ˆRdu2 · ∇(h1 − h2)(µ1 − µ2).

In the Coulomb case, we recognize from (1.25) in the second right-hand side term the diver-gence of the stress-energy tensor [h1 − h2, h1 − h2], hence

∂tH((µ1, u1), (µ2, u2))

= −2ˆRdµ1∇u2 : (u1 − u2)⊗ (u1 − u2)− 1

cd

ˆRdu2 · div [h1 − h2, h1 − h2]

≤ C‖∇u2‖L∞ˆRd|u1 − u2|2µ1 + C

cd‖∇u2‖L∞

ˆRd|∇(h1 − h2)|2

= C‖∇u2‖L∞H((µ1, u1), (µ2, u2)),

and the result follows by Gronwall’s lemma. In the Riesz case, replacing (1.25) by (3.2) andby (4.1)–(4.2), the same conclusion follows. �

Taking inspiration from the above calculations for the weak-strong stability principle, wenow turn to the proof of Theorem 2.

Proof of Theorem 2. Let h := g ∗ µ. We decompose

∂tHN (ZN , (µ, u)) = 2NN∑i=1

(u(xi)− vi) ·((∂tu)(xi) + xi · ∇u(xi)− vi

)+2

∑i 6=j

xi·∇g(xi−xj)+2N2ˆRdh∂tµ−2N

N∑i=1

ˆRdxi·∇g(xi−y) dµ(y)−2N

N∑i=1

ˆRd

g(xi−·) ∂tµ.


Inserting equations (A.1) and (A.3), we obtain

∂tHN (ZN , (µ, u))

= 2NN∑i=1

(u(xi)− vi) ·(− (u · ∇u)(xi)−∇h(xi) + vi · ∇u(xi) + 1

N

∑j:j 6=i∇g(xi − xj)

)+ 2

∑i 6=j

vi · ∇g(xi − xj) + 2N2ˆRd∇h · uµ

− 2NN∑i=1

vi · ∇h(xi)− 2NN∑i=1

ˆRd∇g(· − xi) · u dµ,

and hence, after straightforward simplifications,

∂tHN (ZN , (µ, u)) = −2NN∑i=1∇u(xi) : (u(xi)− vi)⊗ (u(xi)− vi)

+N2¨

Rd×Rd\4(u(x)− u(y)) · ∇g(x− y) d(µN − µ)(x) d(µN − µ)(y).

Applying Proposition 1.1 to estimate the second right-hand side term, we are led to

∂tHN (ZN , (µ, u))

≤ C‖∇u‖L∞(HN (ZN , (µ, u)) + (1 + ‖µ‖L∞)N1+ s

d +N2− s+12 +

(Nd logN

)1(1.4)

)

+ C

(1 + ‖µ‖Cσ)‖∇u‖L∞N2− 1

d + ‖u‖W 1,∞‖µ‖CσN1+ s+1−σd , if s ≥ d− 1,

(1 + ‖µ‖L∞)‖∇u‖L∞N2− 1d + ‖u‖L∞‖µ‖L∞N1+ s+1

d

+‖∇u‖L∞‖µ‖L∞N1+ sd , if s < d− 1,

and the conclusion follows by Gronwall’s lemma. �

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Mitia DuerinckxUniversite Paris-Saclay, CNRS, Laboratoire de Mathematiques d’Orsay, 91405 Orsay, France& Universite Libre de Bruxelles, Departement de Mathematique, 1050 Brussels, Belgium.Email: [email protected] SerfatyCourant Institute, New York University, 251 Mercer street, New York, NY 10012, USA.Email: [email protected]

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